Existence of Periodic Solutions for a Class of Second Order Ordinary Differential Equations
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Existence of Periodic Solutions for a Class of Second Order Ordinary Differential Equations Antonio Garcia1 · Jaume Llibre2
Received: 3 July 2019 / Accepted: 21 October 2019 © Springer Nature B.V. 2019
Abstract We provide sufficient conditions for the existence of a periodic solution for a class of second order differential equations of the form x¨ + g(x) = εf (t, x, x, ˙ ε), where ε is a small parameter. Keywords Periodic orbit · Second-order differential equation · Averaging theory Mathematics Subject Classification (2010) 37G15 · 37C80 · 37C30
1 Introduction and Statement of the Results The second order differential equations of the form x¨ + g(x) = εf (t, x, x, ˙ ε), have been studied by many authors because they have many applications, see for instance [2, 3, 5, 8–12, 15, 17, 19]. Two of the main families studied are the Duffing equations see [6, 7], and the forced pendulum, see the nice survey [14] and the references quoted therein. The aim of this work is to study periodic solutions of the second order differential equation x¨ + g(x) = μ2n+1 p(t) + μ4n+1 q(t, x, x, ˙ μ), where n is a positive integer, μ is a small parameter, and the functions g(x) = x + x 2n+1 b + xh(x) ,
B J. Llibre
[email protected] A. Garcia [email protected]
1
Departamento de Matemáticas, UAM-Iztapalapa, 09340 Iztapalapa, Mexico City, Mexico
2
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
(1)
A. Garcia, J. Llibre
and h(x) are smooth, b = 0, p(t) and q(t, x, y, μ) are smooth and periodic with period 2π in the variable t . Let Γ (x) be the Gamma function, see for more details [1], and let α and β be the first Fourier coefficients of the periodic function p(t), i.e. 1 α= π
2π
1 β= π
p(t) cos t dt, 0
2π
p(t) sin t dt. 0
Now our main result is the following. Theorem 1 If αβ = 0 then for μ = 0 sufficiently small the differential equation (1) has a 2π -periodic solution x(t, μ) such that x(0, μ) = π
1 4n+2
Γ (n + 2) 2bΓ (n + 32 )
1 2 −n 2n+1 β α 2 +1 + O μ2n . α
Theorem 1 is proved in Sect. 3, where we use the averaging theory for computing periodic solutions, see Sect. 2 for a summary of the results on this theory that we shall need.
2 The Averaging Theory We want to study the T -periodic solutions of the periodic differential systems of the form x = F0 (t, x) + εF1 (t, x) + o(ε),
(2)
with ε > 0 sufficiently small, where F0 , F1 : R × Ω → Rn and F2 : R × Ω × (−ε0 , ε0 ) → Rn are C 2 functions, T -periodic in the variable t , and Ω is an open subset of Rn . Let x(t, z, ε) be the solution of the differential system (2) such that x(0, z, ε) = z. Suppose that the unperturbed system x = F0 (t, x), has an open set V with V ⊂ Ω such that for each z ∈ V , x(t, z, 0) is T -periodic. Let y be an n × n matrix, and consider the first order variational equation y = Dx F0 t, x(t, z, 0) y,
(3)
(4)
of the unperturbed system (3) on the periodic solution x(t, z, 0). Let Mz (t) be the fundamental matrix of the linear differential system
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